zbMATH — the first resource for mathematics

A complete discrimination system for polynomials with complex coefficients and its automatic generation. (English) Zbl 0949.12002
Write the polynomial \(f(x)\) as \(f_1(x)+ if_2(x)\), where \(f_1\) and \(f_2\) have real coefficients. The authors give a method of determining the numbers of real zeros, pairs of complex conjugate zeros, and complex zeros with no conjugate (which can arise if \(f_2\) has some non-zero coefficients) of \(f\).
The method is based on constructing sequences of numbers starting from the signs of determinants of subresultants of various polynomials and their derivatives. It has been implemented in Maple, and some run times are given. The authors also consider dealing with polynomials with literal coefficients and the problem of investigating the positive definiteness of polynomials in two variables.
Proofs are not given, but are referred to from papers inaccessible to this reviewer. Examples, for reasons of space, are not given in full.

12E05 Polynomials in general fields (irreducibility, etc.)
12Y05 Computational aspects of field theory and polynomials (MSC2010)
68W30 Symbolic computation and algebraic computation
Full Text: DOI
[1] Gao Xiaoshan, The discriminant systems of unary equation and their computation,MM-preprints, 1987, 1: 13.
[2] Wu Wenda, A note to the discriminant systems for the unary equations,MM-preprints, 1989, 3: 33.
[3] Yang Lu, Hou Xiaorong, Zeng Zhenbing, A complete discrimination system for polynomials,Science in China, Ser. E, 1996, 39(6): 628. · Zbl 0866.68104
[4] Yang Lu, Zhang Jingzhong, Hou Xiaorong,Non-linear Equation Systems and Automated Theorem Proving (in Chinese), Shanghai: Shanghai Press of Science, Technology and Education, 1996.
[5] Robinson, R. M., Some definite polynomial which are not sums of squares of real polynomials,Notice Amer. Math. Soc., 1969, 16: 554.
[6] Wang Dongming, A decision method for definite polynomial,MM-preprints, 1987, 2: 68.
[7] Zhang Jingzhong, Yang Lu, Hou Xiaorong, The subresultant method for automated theorem proving,J. Sys. Sci. & Math. Scis. (in Chinese), 1995, 15(1): 10. · Zbl 0839.68090
[8] Heck, A.,Introduction to Maple, Berlin: Springer-Verlag, 1993. · Zbl 0779.65001
[9] Collins, G. E., Quantifier elimination for real closed fields by cylindrical algebraic decomposition,LNCS, 1975, 33: 134. · Zbl 0318.02051
[10] Liang Songxin, Li Chuanzhong, Determination on positive semi-definiteness of binary polynomials over rationals,Computer Applications (in Chinese), 1998, 18(3): 28.
[11] Winkler, F.,Polynomial Algorithms in Computer Algebra, New York: Springer Wien, 1996. · Zbl 0853.12003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.